 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-4/16c/b/vi/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

ω = Ω = g k = (\frac{π g 5}{L})^{1/2} .

At this value, the initially quiescent resonant solution is

H (t) = - \frac{k p _{0}}{2 ρ Ω}, t sin (Ω t),

F (t) = - \frac{p _{0}}{2 ρ Ω} (sin (Ω t) + Ω t cos (Ω t)) .

Both contain a term growing linearly in time. This is resonance: the pressure forcing repeatedly supplies energy in phase with the box's

(1, 2)

standing-wave mode. In the ideal inviscid linear model there is no damping or nonlinear saturation, so its amplitude is unbounded. This is the resonance of a pressure-forced rectangular surface-gravity mode.

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